80. Find all values of c that satisfy the conclusion of Cauchy...

Question

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let F of X is equal to x the power of 3 divided by 2 + X and G of X is equal to x the power of 2. On the intervals of 0.2, by Cauchy's mean value theorem, find all values C is an element of 0.2. OK. So it appears for this particular problem we're asked to take all the information that is provided to us, which we're told what the function F of X is and what the function G of X is, and we're told the interval, which is parentheses open interval 0.2. We know that it's an open interval because we're, we see parentheses now if it were in square brackets, that would mean that it's a closed interval. So essentially, we're going to be using Cauchy's mean value theorem. This is what we're ultimately trying to solve for is we're going to use Cauchy's mean value theorem to find all values of C as an element of the open interval 0.2. That's what that that weird symbol that looks like an E, that means that it's an element of. So, With that said, now that we know what we're ultimately trying to solve for, our first step that we need to take in order to solve this particular prompt is we need to state what Cauchy's mean value theorem formula is. What is Cauchi's mean value theorem. So as we should recall, we need to note that there exists C is an element of the open interval 0.2. With F C divided by G C is equal to F of B minus F of A all divided by G of B minus G of A. OK, so now that we've stated what Cauchy's mean value theorem is, our second step is we need to recall and use the quotient rule to find F of X. So once again, this is our second step. We need to recall and use the quotient rule to find what F of X is. So F of X will be equal to, so once again we're solving for F of X. So F of X is equal to 3 x 2 multiplied by parentheses 2 plus X minus X the power of 3. All divided by, so once again, all divided by parentheses 2 + X to the power of 2. So in parentheses 2 + X to the power of 2. And this will be equal to, when we write it in its most simple form, so if we use a little bit of algebra to simplify, our most simple form will be equal to 6 X2, so 6 X2 plus 2x the power of 3 divided by parentheses 2 plus x ins to the power of 2. So 2 + X in parentheses to the power of 2. So therefore, thus we can state that F of C will be equal to 6C2 + 2C the power of 3 divided by parentheses of 2 + C to the power of 2. So, we now for our third step is we need to solve for G of X. So we're solving for G of X now, and we will find that G of X will be equal to 2 X. Thus we can go ahead and write that G of C will be equal to 2 C. So once again G of X is equal to 2 X, and with that knowledge we can safely say that G of C will be equal to 2 C. So our fourth step now is we need to evaluate our end points, so we need to find F of 0. And once again we're plugging in a value of 0 into our expression for our value of X, and when we do, we will find that it's equal to 0 to the power of 3 divided by 2 + 0, which is simply equal to 0. And we need to find that F of 2, so now we're plugging in a value of 2 in for X, and we will find that this is equal to 2 to the power of 3 divided by 2 + 2, which is equal to 8 divided by 4, which is simply equal to 2. Now we need to do a similar methodology for G of 0 and G of 0, which once again we're plugging in a value of X into our function of G of X. So G of 0 is going to be equal to 0 to the power of 2, which is simply equal to 0, and G of 2, so now we're putting in a value of 2, and we will find that 2 of 2 is simply equal to 4. Fantastic. Moving right along here. We now need to substitute our values into Cauchy's mean value theorem formula, and when we do, we will find that we can go ahead and write that 6 squad, so 6C2 plus 2C to the power of 3 divided by parentheses 2 + C to the power of 2. All divided by 2C, which will be simply equal to 2 minus 0, divided by 4 minus 0. And when we do a little bit of simplifying using some algebra, you will find that when we simplify, you will get 3 C plus C to the power of 2. All divided by 4 + 4 C plus C squad, which is simply equal to 1/2 or 1 divided by 2 because 2 minus 0 is +24 minus 0 is 4, and then that reduces to 1/2 or 1 divided by 2. So we need to keep simplifying, and when we do, we will get 6 C plus 2C the power of 2 is equal to 4 + 4C + C to the power of 2. Or C square. And when we do a little bit of rearranging and we isolate, so we're going to be moving our values over to one side so we can get it to be set equal to 0, we will find that C2 + or C to the power 2 plus 2 C minus 4 is simply equal to 0. And once again, this is when we rearrange our variables, so we get a 0 on one side and then we move all of our variables to another side. Fantastic. So now what we need to do is we need to find the value of C by using the quadratic formula. So we need to recall and use the quadratic formula. So C is equal to -2 plus or minus the square root of 2 of 2, minus 4 multiplied by 1, multiplied by -4. All divided by 2 multiplied by 1. And when we simplify when we solve for C, we will simply find that C is equal to -1 plus or minus the square root of 5. We will also find that C is equal to -1 plus the square root of 5, which is approximately equal to 1.24, and C. is equal to -1 minus the square root of 5. So once again, when we say C is equal to -1 plus or minus the square root of 5. We will find, when we take -1 minus the square root of phi, we will find that it's approximately equal to -3.24. So, we need to note that our correct and final answer means we do not want to have a negative value for our final answer for C. So our correct and final answer is C is equal to -1 plus the square root of phi, and that is our correct and final answer. Hooray, we did it. So the solution in our interval, the open interval 0.2, will be -1 plus the square root of 5. Hooray, we did it. So to quickly recap, going back to review our problem itself, we can safely say without a doubt that our final answer once again is as follows. So our final answer once again without a doubt will be that -1. Plus the square root of 5, and that's it, we've officially solved for this problem. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)

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80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)